A math circle is broadly defined as a semi-formal, sustained enrichment experience that brings mathematics professionals in direct contact with pre-college students and/or their teachers. Circles foster passion and excitement for deep mathematics.

These two demonstration sessions, each directed by an experienced Math Circle leader, offer the opportunity for MATHFEST 2012 attendees to observe and take part in Math Circle experiences, and enjoy the thrill the organic and creative process the conversational style of learning Circles offer. The first is directed towards professional mathematicians as participants, the second towards student as participants. Both are for all to witness.

These sessions are supported by SIGMAA for Math Circles for Students and Teachers (SIGMAA MCST). Seeing a circle in action, we believe, is the best way to generate enthusiasm to start one of your own. Come see why!

What is the role of visual information in mathematics? Can diagrams justify inferences in traditional, verbal proofs? Can pictures, or diagrams, be proofs on their own? There is much disagreement on these issues among both mathematicians and philosophers; part of the reason for the disagreement is confusion. The aim of this talk is to clarify some of the philosophical issues underlying disputes over the role of visual information in proofs. Diagrams can be highly convincing, useful for explaining, they can efficiently depict mathematical information. But that does not mean they are proofs. This talk will appeal to some general considerations in epistemology to explain the view that pictures fall short of being genuine mathematical proofs. But proofs are just one tool in the mathematician's toolbox; we will also aim to clarify why pictures can be so useful, convincing, and even justifying!

Both intuitive and formal aspects of limit concepts have proven difficult for undergraduate students in lower-division mathematics and introductory proof courses. Our research has investigated the cognitive challenges these students encounter while developing and formalizing a robust understanding of a variety of limit concepts. We also seek to identify particular solutions, general characteristics of students’ inquiry, and instructional supports which foster effective and lasting resolutions to these challenges. The two sessions of our workshop will focus on the translation of our research results to instruction in undergraduate mathematics courses.

During the first session, we will present an instructional cycle that supports students' reinvention of formal definitions for sequence convergence, series convergence, and pointwise convergence. Workshop participants will work through some of the mathematical tasks for themselves, consider the theory and research results that guided the creation of the tasks, watch video of students working on these tasks, and discuss possible implications for both instruction and instructional design in this and other areas.

The second session will focus on the role of the instructor in supporting students’ reinvention activity. We will address issues of developing critical reasoning and argumentation, fostering students’ ownership of their mathematical activity and its products, and bridging the gap between students’ emerging formal reasoning and their ability to express these ideas in verbal and written mathematical language. We will conclude with an investigation of the nature of students’ intuitive limit concepts prior to the reinvention activity and the implications for instruction in introductory calculus courses.

Engaging High School Students in Research Experiences
Saturday, August 4, 5:30-6:30 p.m., Hall of Ideas F

Organizers:

During June 2012, SIGMAA TAHSM sponsored a PREP workshop for faculty at high schools, community colleges, and small liberal arts colleges to support them in engaging their students in a mathematics research experience during the 2012-13 school year. Following the business meeting, and with pizza in hand, participants from the June PREP session will discuss some of the mathematical problems considered during the summer, give an overview of the program as it is to be implemented in schools next year, and lead an open forum on future activities that engage high school teachers with mathematics and the MAA.

Daniel Teague, North Carolina School of Science and Mathematics

Mathematics Instruction Using the Web: WEB SIGMAA

Discussion Topic: Online Technology in Mathematics Education
An open discussion with a panel and members of the audience relating to the following points:

Augmenting the traditional classroom with online resources

Emerging technologies in the not so distant future. What will we see in the next 2-10 years?